国际数学日特辑｜数学与希望：对话SIMIS教授

3月14日，不仅是我们熟知的“圆周率日（Pi Day）”，也是由联合国教科文组织设立的“国际数学日”（International Day of Mathematics）。今年，官方赋予了这一天一个温暖而充满力量的主题——“数学与希望”（Mathematics and Hope）。

什么是数学中的希望？对于科研工作者而言，希望或许是在长久枯竭的思路中突然闪现的灵光，是对那些看似无解的宇宙与数字之谜的坚持，更是通过跨学科的交叉融合去重塑人类认知的无限可能。

在这个致敬人类智慧结晶的特殊日子里，上海数学与交叉学科研究院（SIMIS）带您走进两位优秀科研人员的世界——莫仲鹏教授与Mauricio Andrés Romo Jorquera 副教授。

从“代数数论”的前沿激荡，到“数学物理”的深邃幽光，让我们一起聆听他们是如何在科研的“艰难时刻”中孕育希望，破局而出。

Q：在您的研究过程中，做数学研究常常会“卡住”，您是如何走出困境的？Have you faced any challenges in your research so far, and how did you get through these “tough moments”?

A: 做数学研究的确经常会“卡住”，毕竟我们尝试的是建构新的定理。与同行讨论，或者查阅文献去了解他人已经实践过的思路，是走出困境的有效方法。只要你不轻易放弃，那么即使或许不能完整解决原来遇到的问题，也常常能有新的、有益的收获。

It is indeed true that we frequently “got stuck” in research; after all we are aiming to establish new theorems. Discussing with colleagues or reading papers to see what others have done is a good way to get through the tough moments. If one does not give up, then time and again one could come up with something useful, even if one cannot solve the original problem in entirety.

Q：您认为目前重塑您研究领域的主要驱动力是什么？未来有哪些新挑战或机遇？
What is the main power reshaping your research area? What new challenges or opportunities do you foresee?

A：当下，人工智能的确正在成为研究中的有力工具，它们能在短时间内梳理大量文献。但就代数数论领域而言，我们已经见证了许多著名难题在过去几十年的突破，如费马大定理、塞尔猜想、佐藤–泰特猜想、朗兰兹–谢尔斯塔德基本引理等。剩下的核心问题都十分艰深，可能需要全新的思想才能取得突破。

AI is indeed providing a very useful tool for research today, as it can summarize a large part of the research literature quickly. However, in algebraic number theory we have witnessed many major breakthroughs—Fermat’s Last Theorem, Serre’s conjecture, Sato–Tate, the Langlands–Shastry Fundamental Lemma. The remaining core problems are extremely difficult, and significant new ideas seem necessary for future progress.

寄语：视野要开阔，即使在研究生涯中保持专精也至关重要。现在许多重要成果都是通过多领域方法的交叉融合完成的，因此，找到能与你知识互补、强强联合的合作伙伴十分关键。
Have a broad knowledge base, even though specialization is ultimately necessary. Much good research today comes from combining methods across areas, so finding collaborators whose strengths complement your own is very important.

Q：在您的研究过程中，是否遇到过挑战？您是如何度过这些“艰难时刻”的？Have you faced any challenges in your research so far, and how did you get through these “tough moments”?

A：当然，挑战几乎贯穿科研生涯的每一个阶段。研究过程中，人们常常会遇到不知该如何推进的问题：不确定该进行哪一种计算，思路似乎逐渐枯竭，甚至连如何恰当地表述一个问题都变得困难。
面对困难是科研的重要阶段之一，也正是在这些阶段，真正的进展才有可能发生。如果一切始终顺利，那或许意味着研究并未触及最前沿的问题。对我而言，应对这些“艰难时刻”的最好方法，是尝试将当前的问题简化，甚至将其简化到几乎显而易见的程度，然后再逐步引入复杂性。
Many challenges arise at all stages of a scientific career. One is often confronted with situations where it is unclear how to proceed—what computation to carry out, how to formulate a problem correctly, or even how to generate new ideas when one seems to have run out of them.
Facing such difficulties is an essential part of research, and it is precisely during these moments that genuine progress can occur. If everything always went smoothly, one might not truly be engaging in cutting-edge research. For me, the best way to get through tough moments is to try to reformulate the problem in a simpler form—sometimes even to the point where it becomes trivial—and then gradually reintroduce complexity step by step.

Q：您认为当前重塑您研究领域的主要力量是什么？未来是否存在新的挑战或机遇？In your view, what is the main power reshaping your research area today? What challenges or opportunities do you foresee?

A：数学物理是一个极为广阔的领域，其中一些重要方向包括镜像对称、拓扑弦理论以及超对称规范场论等。当前，仍有大量纯数学与理论物理中的研究方向，尚未在一个清晰而严谨的交叉界面中得到充分探索。
例如，超弦理论中提出的 AdS/CFT 对应这一革命性猜想，虽然已有二十多年的历史，但其中的若干方面直到最近才逐渐进入纯数学的研究视野。类似地，BPS 态计数作为超对称规范场论的核心内容之一，在提出三十多年后，依然不断为我们产出新的发现与惊喜。
因此，我认为数学物理领域仍然充满开放问题与突破机遇。我尤其建议对这一方向感兴趣的青年数学家，多学习理论物理，并尝试按照物理学家理解和使用这些理论的方式去接近它们。维格纳在 1960 年曾提出“数学在自然科学中不可思议的有效性”，而今天，我们或许也应当承认理论物理在纯数学中同样展现出了不可思议的有效性。
Mathematical physics is a vast field, with major areas including mirror symmetry, topological string theory, and supersymmetric gauge theories, among many others. There remain numerous directions in pure mathematics and theoretical physics that are still awaiting exploration within a well-defined interface between the two.
For instance, certain aspects of the AdS/CFT correspondence—a revolutionary conjecture in string theory proposed over twenty years ago—have only recently begun to influence pure mathematics. Similarly, the physics and mathematics of BPS state counting, a cornerstone of supersymmetric gauge theories, continue to yield new insights more than thirty years after their inception.
I would therefore say that the field offers many open problems and opportunities for breakthroughs. I strongly encourage young mathematicians interested in this area to learn more theoretical physics, in the way physicists themselves understand it. As Eugene Wigner famously spoke of the “unreasonable effectiveness of mathematics in the natural sciences,” we may now also recognize the unreasonable effectiveness of theoretical physics in pure mathematics.

寄语

我建议中国的青年学者始终关注自身研究方向周边的发展，即使这些领域在初看时似乎并不相关。例如，从事代数几何研究的学者，若关注超对称规范场论的相关成果，往往能获得新的视角与启发。尽管最初你可能难以理解这些联系，但它们有时会带来意想不到的充实收获。
对于学生而言，我尤其建议你们尽可能多地参加学术报告，并主动接触不同方向的研究。开始时，这些内容或许难以理解，甚至会让你感到迷茫，但随着学术积累的加深，你的视角会越发清晰。请相信这些努力必定会给你的未来带来影响深远的回报。

For young scholars in China, I would recommend keeping an eye on developments in areas adjacent to their own research fields, even if they initially seem unrelated. For example, results from supersymmetric gauge theories may provide new ideas or unexpected insights for researchers in algebraic geometry. Although the connections may not be clear at first, they can sometimes lead to remarkably fruitful breakthroughs.
For students, I strongly encourage attending as many talks as possible and exploring topics that spark curiosity. In the beginning, these talks may be difficult to follow or even confusing, but over time, as one progresses academically, a clearer picture will emerge. This effort will undoubtedly prove valuable in the long run.

在国际数学日即将来临之际，祝愿所有热爱数学、不断探索的学者与学子们永远怀揣希望；愿你们的灵感如π一般，无限延展，生生不息！